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States

Lattice states in Schwinger.jl are represented by the abstract type SchwingerState, with four concrete types:

  • BasisState: a state specified by the eigenvalues of occupation operators $\chi^\dagger_{n,\alpha}\chi_{n,\alpha}$ and $L_0$
  • EDState: a linear combination of BasisStates
  • ITensorState: a matrix product state using ITensorMPS.jl
  • MPSKitState: a matrix product state using MPSKit.jl

Given a state, we can find the expectation values of the occupation operators and electric field operators:

using Schwinger
lat = Lattice(6; F = 1, a = 10) # towards the lattice strong coupling limit ga -> infty
gs = groundstate(Hamiltonian(lat; backend=:ED))

occupations(gs), electricfields(gs)
([0.9996009569748595; 0.0007976086592019167; … ; 0.9992023913407977; 0.0003990430251401046;;], [-0.0003990430251404886; 0.00039856563406142807; … ; -0.00039904302514120744; -1.1028504498522551e-15;;])

We can also evaluate the entanglement entropies of each bisection of the lattice:

using Schwinger
lat = Lattice(20; F = 1, )
gs = groundstate(Hamiltonian(lat; backend=:ITensors))
entanglements(gs)
19-element Vector{Float64}:
 0.5892661594332941
 0.39444995102389885
 0.516866301480542
 0.45373049909707325
 0.489916146643204
 0.4701048867712342
 0.4812980874019193
 0.4748933776117914
 0.4788497809441128
 0.475938063085563
 0.47884978094398795
 0.4748933776117382
 0.48129808740185354
 0.47010488677116635
 0.4899161466431349
 0.45373049909698904
 0.5168663014802899
 0.39444995102369096
 0.5892661594332924

When using an infinite lattice with MPSKit, we can compute these quantities in the thermodynamic limit.

using Schwinger
lat = Lattice(Inf; F = 1, )
gs = groundstate(Hamiltonian(lat; backend=:MPSKit))
entanglements(gs)
2-element Vector{Float64}:
 0.47727450642310904
 0.47727450642638825

The two numbers here correspond to the two possible bisections of the two-site unit cell (corresponding to an odd and an site from the middle of a large finite lattice).

Several other useful functions are detailed below.

Schwinger.EDState — Type

EDState(hamiltonian, coeffs)

A Schwinger model state represented as a linear combination of basis states.

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Schwinger.occupation — Function

occupation(state, site)

Return the expectations of χ†χ operators of each flavor on a given site.

Arguments

  • state::ITensorState: Schwinger model state.
  • site::Int: the lattice site.
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occupation(state, site)

Return the expectations of χ†χ operators of each flavor on a given site.

Arguments

  • state::MPSKitState: Schwinger model state.
  • site::Int: the lattice site.
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occupation(state, site)

Return the expectations of χ†χ operators of each flavor on a given site.

Arguments

  • state::BasisState: Schwinger model basis state.
  • site::Int: the lattice site.
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occupation(state, site)

Return the expectations of χ†χ operators of each flavor on a given site.

Arguments

  • state::EDState: Schwinger model basis state.
  • site::Int: the lattice site.
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Schwinger.occupations — Function

occupations(state)

Return an NxF matrix of the expectations of χ†χ operators on each site.

Arguments

  • state::ITensorState: Schwinger model state.
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occupations(state)

Return an NxF matrix of the expectations of χ†χ operators on each site.

Arguments

  • state::MPSKitState: Schwinger model state.
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occupations(state)

Return an NxF matrix of the expectations of χ†χ operators on each site.

Arguments

  • state::BasisState: Schwinger model basis state.
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occupations(state)

Return an NxF matrix of the expectations of χ†χ operators on each site.

Arguments

  • state::EDState: Schwinger model basis state.
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Schwinger.charge — Function

charge(state, site)

Return the expectation value of the charge operator on site site.

Arguments

  • state::SchwingerState: Schwinger model state.
  • site::Int: site.
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Schwinger.charges — Function

charges(state)

Return a list of the expectations of Q operators on each site.

Arguments

  • state::SchwingerState: Schwinger model state.
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Schwinger.electricfield — Function

electricfield(state, link)

Return the expectation of (L + θ/2π) on the link link.

Arguments

  • state::SchwingerState: Schwinger model state.
  • link::Int: link.
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Schwinger.electricfields — Function

electricfields(state)

Return a list of the expectations of (L + θ/2π) operators on each link.

Arguments

  • state::SchwingerState: Schwinger model state.
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Schwinger.entanglement — Function

entanglement(state, bisection)

Return the von Neumann entanglement entropy -tr(ρₐ log(ρₐ)), where a is the subsystem of sites 1..bisection

Arguments

  • state::EDState: Schwinger model state.
  • bisection::Int: bisection index.
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entanglement(state, bisection)

Return the von Neumann entanglement entropy -tr(ρₐ log(ρₐ)), where a is the subsystem of sites 1..bisection

Arguments

  • state::ITensorState: Schwinger model state.
  • bisection::Int: bisection index.
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entanglement(state, bisection)

Return the von Neumann entanglement entropy -tr(ρₐ log(ρₐ)), where a is the subsystem of sites 1..bisection

Arguments

  • state::MPSKitState: Schwinger model state.
  • bisection::Int: bisection index.
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Schwinger.entanglements — Function

entanglements(state)

Return a list of the von Neumann entanglement entropies for each bisection of the lattice.

Arguments

  • state::SchwingerState: Schwinger model state.
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Schwinger.energy — Function

energy(state)

Return the expectation value of the Hamiltonian.

Arguments

  • state::SchwingerState: Schwinger model state (ED, MPS, or MPSKit).
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energy(state)

Return the expectation value of the Hamiltonian.

Arguments

  • state::BasisState: Schwinger model basis state.
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Schwinger.L₀ — Function

L₀(state)

Return the expectation value of L₀.

Arguments

  • state::SchwingerState: Schwinger model state.
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Schwinger.scalar — Function

scalar(state)

Return the expectation value of the scalar condensate, ⟨H_mass⟩/L.

Arguments

  • state::SchwingerState: Schwinger model state.
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Schwinger.scalardensity — Function

scalardensity(state, site)

Return the scalar density at site site.

Arguments

  • state::SchwingerState: Schwinger model state.
  • site::Int: site.
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Schwinger.scalardensities — Function

scalardensities(state)

Return the list of scalar densities of state on sites 1 through N.

Arguments

  • state::SchwingerState: Schwinger model state.
  • site::Int: site.
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Schwinger.pseudoscalar — Function

pseudoscalar(state)

Return the expectation value of the pseudoscalar condensate, ⟨H_hoppingmass⟩/L.

Arguments

  • state::SchwingerState: Schwinger model state.
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Schwinger.pseudoscalardensity — Function

pseudoscalardensity(state, n)

Return the pseudoscalar density at site n.

Arguments

  • state::SchwingerState: Schwinger model state.
  • n::Int: site.
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Schwinger.pseudoscalardensities — Function

pseudoscalardensities(state)

Return the list of pseudoscalar densities of state on sites 1 through N.

Arguments

  • state::SchwingerState: Schwinger model state.
  • site::Int: site.
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